#117311

order	512
nilpotency class	3
isoclinism	isoclinic class of #117305
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

754 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,5,8,6,10,4,9)(3,11,14,16,12,15,13,7)(17,18,22,26)(19,27,30,32)(20,24,21,25)(23,29,31,28);(1,3,4,13,6,12,5,14)(2,7,8,11,10,16,9,15)(17,19,20,28,22,30,21,29)(18,23,25,27,26,31,24,32);(2,8)(3,12)(4,5)(7,15)(9,10)(11,16)(18,24)(19,28)(20,21)(23,31)(25,26)(29,30)

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,e^b=eh,f^b=fhi,d^c=dh,e^c=eh,f^c=fi,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24507);